\(\int x^3 (c+a^2 c x^2)^{5/2} \arctan (a x)^3 \, dx\) [428]

Optimal result

Integrand size = 24, antiderivative size = 798 \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\frac {85 c^2 x \sqrt {c+a^2 c x^2}}{12096 a^3}-\frac {c^2 x^3 \sqrt {c+a^2 c x^2}}{240 a}-\frac {1}{504} a c^2 x^5 \sqrt {c+a^2 c x^2}-\frac {6157 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{60480 a^4}-\frac {47 c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{30240 a^2}+\frac {67 c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)}{2520}+\frac {1}{84} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {47 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{896 a^3}-\frac {205 c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{4032 a}-\frac {103 a c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{1008}-\frac {1}{24} a^3 c^2 x^7 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {115 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{1344 a^4 \sqrt {c+a^2 c x^2}}-\frac {2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{63 a^4}+\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{63 a^2}+\frac {5}{21} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {19}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1433 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{15120 a^4}+\frac {115 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{1344 a^4 \sqrt {c+a^2 c x^2}}-\frac {115 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{1344 a^4 \sqrt {c+a^2 c x^2}}-\frac {115 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{1344 a^4 \sqrt {c+a^2 c x^2}}+\frac {115 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{1344 a^4 \sqrt {c+a^2 c x^2}} \]

[Out]

1433/15120*c^(5/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))/a^4-115/1344*I*c^3*arctan((1+I*a*x)/(a^2*x^2+1)^(1
/2))*arctan(a*x)^2*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-115/1344*I*c^3*arctan(a*x)*polylog(2,I*(1+I*a*x)/
(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)+115/1344*I*c^3*arctan(a*x)*polylog(2,-I*(1+I*a*x)
/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-115/1344*c^3*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^
(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)+115/1344*c^3*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^
2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)+85/12096*c^2*x*(a^2*c*x^2+c)^(1/2)/a^3-1/240*c^2*x^3*(a^2*c*x^2+c)^(1/2)/a-
1/504*a*c^2*x^5*(a^2*c*x^2+c)^(1/2)-6157/60480*c^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^4-47/30240*c^2*x^2*arctan
(a*x)*(a^2*c*x^2+c)^(1/2)/a^2+67/2520*c^2*x^4*arctan(a*x)*(a^2*c*x^2+c)^(1/2)+1/84*a^2*c^2*x^6*arctan(a*x)*(a^
2*c*x^2+c)^(1/2)+47/896*c^2*x*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^3-205/4032*c^2*x^3*arctan(a*x)^2*(a^2*c*x^2+
c)^(1/2)/a-103/1008*a*c^2*x^5*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)-1/24*a^3*c^2*x^7*arctan(a*x)^2*(a^2*c*x^2+c)^(
1/2)-2/63*c^2*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/a^4+1/63*c^2*x^2*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/a^2+5/21*c^
2*x^4*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)+19/63*a^2*c^2*x^6*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)+1/9*a^4*c^2*x^8*ar
ctan(a*x)^3*(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 13.85 (sec) , antiderivative size = 798, normalized size of antiderivative = 1.00, number of steps used = 547, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5070, 5072, 5050, 223, 212, 5010, 5008, 4266, 2611, 2320, 6724, 327} \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\frac {1}{9} a^4 c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3 x^8-\frac {1}{24} a^3 c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7+\frac {19}{63} a^2 c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3 x^6+\frac {1}{84} a^2 c^2 \sqrt {a^2 c x^2+c} \arctan (a x) x^6-\frac {103 a c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{1008}-\frac {1}{504} a c^2 \sqrt {a^2 c x^2+c} x^5+\frac {5}{21} c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3 x^4+\frac {67 c^2 \sqrt {a^2 c x^2+c} \arctan (a x) x^4}{2520}-\frac {205 c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4032 a}-\frac {c^2 \sqrt {a^2 c x^2+c} x^3}{240 a}+\frac {c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3 x^2}{63 a^2}-\frac {47 c^2 \sqrt {a^2 c x^2+c} \arctan (a x) x^2}{30240 a^2}+\frac {47 c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2 x}{896 a^3}+\frac {85 c^2 \sqrt {a^2 c x^2+c} x}{12096 a^3}-\frac {2 c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{63 a^4}-\frac {115 i c^3 \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{1344 a^4 \sqrt {a^2 c x^2+c}}-\frac {6157 c^2 \sqrt {a^2 c x^2+c} \arctan (a x)}{60480 a^4}+\frac {1433 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{15120 a^4}+\frac {115 i c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}}-\frac {115 i c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}}-\frac {115 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}}+\frac {115 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}} \]

[In]

Int[x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]

[Out]

(85*c^2*x*Sqrt[c + a^2*c*x^2])/(12096*a^3) - (c^2*x^3*Sqrt[c + a^2*c*x^2])/(240*a) - (a*c^2*x^5*Sqrt[c + a^2*c
*x^2])/504 - (6157*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(60480*a^4) - (47*c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a
*x])/(30240*a^2) + (67*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/2520 + (a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan
[a*x])/84 + (47*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(896*a^3) - (205*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a
*x]^2)/(4032*a) - (103*a*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/1008 - (a^3*c^2*x^7*Sqrt[c + a^2*c*x^2]*Ar
cTan[a*x]^2)/24 - (((115*I)/1344)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^4*Sqrt[c +
a^2*c*x^2]) - (2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(63*a^4) + (c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3
)/(63*a^2) + (5*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/21 + (19*a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x
]^3)/63 + (a^4*c^2*x^8*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/9 + (1433*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2
*c*x^2]])/(15120*a^4) + (((115*I)/1344)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/
(a^4*Sqrt[c + a^2*c*x^2]) - (((115*I)/1344)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])
/(a^4*Sqrt[c + a^2*c*x^2]) - (115*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(1344*a^4*Sqrt[c +
a^2*c*x^2]) + (115*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(1344*a^4*Sqrt[c + a^2*c*x^2])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4266

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 5008

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[1/(c*Sqrt[d]), Subst
[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] &
& GtQ[d, 0]

Rule 5010

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*
d] && IGtQ[p, 0] &&  !GtQ[d, 0]

Rule 5050

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(d + e*x^2)^(
q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Dist[b*(p/(2*c*(q + 1))), Int[(d + e*x^2)^q*(a + b*ArcTan[c
*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]

Rule 5070

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Dist[
d, Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Dist[c^2*(d/f^2), Int[(f*x)^(m + 2)*(d + e*
x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] &&
IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))

Rule 5072

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*((a + b*ArcTan[c*x])^p/(c^2*d*m)), x] + (-Dist[b*f*(p/(c*m)), Int[(f*x)^(m - 1
)*((a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Dist[f^2*((m - 1)/(c^2*m)), Int[(f*x)^(m - 2)*((a +
b*ArcTan[c*x])^p/Sqrt[d + e*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && Gt
Q[m, 1]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = c \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx+\left (a^2 c\right ) \int x^5 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx \\ & = c^2 \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx+2 \left (\left (a^2 c^2\right ) \int x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx\right )+\left (a^4 c^2\right ) \int x^7 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx \\ & = c^3 \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {x^5 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^7 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^3\right ) \int \frac {x^5 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^7 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^6 c^3\right ) \int \frac {x^9 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \frac {c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^2}+\frac {1}{5} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {1}{5} \left (4 c^3\right ) \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (2 c^3\right ) \int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {c^3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a}-\frac {1}{5} \left (3 a c^3\right ) \int \frac {x^4 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{7} \left (6 a^2 c^3\right ) \int \frac {x^5 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\frac {1}{5} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {1}{5} \left (4 c^3\right ) \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{5} \left (3 a c^3\right ) \int \frac {x^4 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{7} \left (6 a^2 c^3\right ) \int \frac {x^5 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{7} \left (3 a^3 c^3\right ) \int \frac {x^6 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )-\frac {1}{7} \left (3 a^3 c^3\right ) \int \frac {x^6 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{9} \left (8 a^4 c^3\right ) \int \frac {x^7 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (a^5 c^3\right ) \int \frac {x^8 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3}-\frac {3 c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{20 a}-\frac {1}{14} a c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{24} a^3 c^2 x^7 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^4}+\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{15 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{10} \left (3 c^3\right ) \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (24 c^3\right ) \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {c^3 \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^3}+\frac {\left (2 c^3\right ) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a^3}+\frac {\left (8 c^3\right ) \int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {c^3 \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}+\frac {\left (9 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{20 a}+\frac {\left (4 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{5 a}+\frac {1}{14} \left (5 a c^3\right ) \int \frac {x^4 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (18 a c^3\right ) \int \frac {x^4 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{7} \left (a^2 c^3\right ) \int \frac {x^5 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (-\frac {3 c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{20 a}-\frac {1}{14} a c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {4 c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{15 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{10} \left (3 c^3\right ) \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (24 c^3\right ) \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (8 c^3\right ) \int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (9 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{20 a}+\frac {\left (4 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{5 a}+\frac {1}{14} \left (5 a c^3\right ) \int \frac {x^4 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (18 a c^3\right ) \int \frac {x^4 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{7} \left (a^2 c^3\right ) \int \frac {x^5 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx\right )+\frac {1}{21} \left (16 a^2 c^3\right ) \int \frac {x^5 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{24} \left (7 a^3 c^3\right ) \int \frac {x^6 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{21} \left (8 a^3 c^3\right ) \int \frac {x^6 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{12} \left (a^4 c^3\right ) \int \frac {x^7 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.37 (sec) , antiderivative size = 850, normalized size of antiderivative = 1.07 \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\frac {c^2 \sqrt {c+a^2 c x^2} \left (774144 \left (-11 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+10 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+11 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-11 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-11 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+11 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )+256 \left (-16407 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+12788 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+16407 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-16407 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-16407 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+16407 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-16128 \left (1+a^2 x^2\right )^{5/2} \left (\frac {48 a x}{\left (1+a^2 x^2\right )^2}+32 \arctan (a x)^3 (-1+5 \cos (2 \arctan (a x)))+6 \arctan (a x) (25+36 \cos (2 \arctan (a x))+11 \cos (4 \arctan (a x)))+\arctan (a x)^2 (6 \sin (2 \arctan (a x))-33 \sin (4 \arctan (a x)))\right )+576 \left (64 \left (309 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-259 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-309 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+309 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+309 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-309 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )+\left (1+a^2 x^2\right )^{7/2} \left (64 \arctan (a x)^3 (57-28 \cos (2 \arctan (a x))+35 \cos (4 \arctan (a x)))+\frac {8 \arctan (a x) (647+764 \cos (2 \arctan (a x))+309 \cos (4 \arctan (a x)))}{1+a^2 x^2}+4 (101 \sin (2 \arctan (a x))+88 \sin (4 \arctan (a x))+25 \sin (6 \arctan (a x)))-3 \arctan (a x)^2 (211 \sin (2 \arctan (a x))-60 \sin (4 \arctan (a x))+103 \sin (6 \arctan (a x)))\right )\right )-\left (1+a^2 x^2\right )^{9/2} \left (1536 \arctan (a x)^3 (-178+711 \cos (2 \arctan (a x))-126 \cos (4 \arctan (a x))+105 \cos (6 \arctan (a x)))+\frac {8 \arctan (a x) (87630+153529 \cos (2 \arctan (a x))+59266 \cos (4 \arctan (a x))+16407 \cos (6 \arctan (a x)))}{1+a^2 x^2}+74932 \sin (2 \arctan (a x))+77908 \sin (4 \arctan (a x))+36612 \sin (6 \arctan (a x))+3 \arctan (a x)^2 (13074 \sin (2 \arctan (a x))-26742 \sin (4 \arctan (a x))+6362 \sin (6 \arctan (a x))-5469 \sin (8 \arctan (a x)))+7238 \sin (8 \arctan (a x))\right )\right )}{15482880 a^4 \sqrt {1+a^2 x^2}} \]

[In]

Integrate[x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]

[Out]

(c^2*Sqrt[c + a^2*c*x^2]*(774144*((-11*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 10*ArcTanh[(a*x)/Sqrt[1 +
a^2*x^2]] + (11*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (11*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTa
n[a*x])] - 11*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 11*PolyLog[3, I*E^(I*ArcTan[a*x])]) + 256*((-16407*I)*ArcTa
n[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 12788*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (16407*I)*ArcTan[a*x]*PolyLog[2,
(-I)*E^(I*ArcTan[a*x])] - (16407*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 16407*PolyLog[3, (-I)*E^(I*
ArcTan[a*x])] + 16407*PolyLog[3, I*E^(I*ArcTan[a*x])]) - 16128*(1 + a^2*x^2)^(5/2)*((48*a*x)/(1 + a^2*x^2)^2 +
32*ArcTan[a*x]^3*(-1 + 5*Cos[2*ArcTan[a*x]]) + 6*ArcTan[a*x]*(25 + 36*Cos[2*ArcTan[a*x]] + 11*Cos[4*ArcTan[a*
x]]) + ArcTan[a*x]^2*(6*Sin[2*ArcTan[a*x]] - 33*Sin[4*ArcTan[a*x]])) + 576*(64*((309*I)*ArcTan[E^(I*ArcTan[a*x
])]*ArcTan[a*x]^2 - 259*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - (309*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x
])] + (309*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 309*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 309*PolyL
og[3, I*E^(I*ArcTan[a*x])]) + (1 + a^2*x^2)^(7/2)*(64*ArcTan[a*x]^3*(57 - 28*Cos[2*ArcTan[a*x]] + 35*Cos[4*Arc
Tan[a*x]]) + (8*ArcTan[a*x]*(647 + 764*Cos[2*ArcTan[a*x]] + 309*Cos[4*ArcTan[a*x]]))/(1 + a^2*x^2) + 4*(101*Si
n[2*ArcTan[a*x]] + 88*Sin[4*ArcTan[a*x]] + 25*Sin[6*ArcTan[a*x]]) - 3*ArcTan[a*x]^2*(211*Sin[2*ArcTan[a*x]] -
60*Sin[4*ArcTan[a*x]] + 103*Sin[6*ArcTan[a*x]]))) - (1 + a^2*x^2)^(9/2)*(1536*ArcTan[a*x]^3*(-178 + 711*Cos[2*
ArcTan[a*x]] - 126*Cos[4*ArcTan[a*x]] + 105*Cos[6*ArcTan[a*x]]) + (8*ArcTan[a*x]*(87630 + 153529*Cos[2*ArcTan[
a*x]] + 59266*Cos[4*ArcTan[a*x]] + 16407*Cos[6*ArcTan[a*x]]))/(1 + a^2*x^2) + 74932*Sin[2*ArcTan[a*x]] + 77908
*Sin[4*ArcTan[a*x]] + 36612*Sin[6*ArcTan[a*x]] + 3*ArcTan[a*x]^2*(13074*Sin[2*ArcTan[a*x]] - 26742*Sin[4*ArcTa
n[a*x]] + 6362*Sin[6*ArcTan[a*x]] - 5469*Sin[8*ArcTan[a*x]]) + 7238*Sin[8*ArcTan[a*x]])))/(15482880*a^4*Sqrt[1
+ a^2*x^2])

Maple [A] (verified)

Time = 19.27 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.66

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (13440 \arctan \left (a x \right )^{3} a^{8} x^{8}-5040 \arctan \left (a x \right )^{2} a^{7} x^{7}+36480 \arctan \left (a x \right )^{3} a^{6} x^{6}+1440 a^{6} \arctan \left (a x \right ) x^{6}-12360 a^{5} \arctan \left (a x \right )^{2} x^{5}+28800 a^{4} \arctan \left (a x \right )^{3} x^{4}-240 a^{5} x^{5}+3216 \arctan \left (a x \right ) a^{4} x^{4}-6150 a^{3} \arctan \left (a x \right )^{2} x^{3}+1920 \arctan \left (a x \right )^{3} x^{2} a^{2}-504 a^{3} x^{3}-188 a^{2} \arctan \left (a x \right ) x^{2}+6345 a \arctan \left (a x \right )^{2} x -3840 \arctan \left (a x \right )^{3}+850 a x -12314 \arctan \left (a x \right )\right )}{120960 a^{4}}+\frac {115 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8064 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {115 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8064 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {1433 i c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{7560 a^{4} \sqrt {a^{2} x^{2}+1}}\)	\(525\)

[In]

int(x^3*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/120960*c^2/a^4*(c*(a*x-I)*(I+a*x))^(1/2)*(13440*arctan(a*x)^3*a^8*x^8-5040*arctan(a*x)^2*a^7*x^7+36480*arcta
n(a*x)^3*a^6*x^6+1440*a^6*arctan(a*x)*x^6-12360*a^5*arctan(a*x)^2*x^5+28800*a^4*arctan(a*x)^3*x^4-240*a^5*x^5+
3216*arctan(a*x)*a^4*x^4-6150*a^3*arctan(a*x)^2*x^3+1920*arctan(a*x)^3*x^2*a^2-504*a^3*x^3-188*a^2*arctan(a*x)
*x^2+6345*a*arctan(a*x)^2*x-3840*arctan(a*x)^3+850*a*x-12314*arctan(a*x))+115/8064*c^2*(c*(a*x-I)*(I+a*x))^(1/
2)*(I*arctan(a*x)^3-3*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)
/(a^2*x^2+1)^(1/2))-6*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^4/(a^2*x^2+1)^(1/2)-115/8064*c^2*(c*(a*x-I)
*(I+a*x))^(1/2)*(I*arctan(a*x)^3-3*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(2
,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^4/(a^2*x^2+1)^(1/2)-1433/7560*I*
c^2/a^4*(c*(a*x-I)*(I+a*x))^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*x^2+1)^(1/2)

Fricas [F]

\[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3} \arctan \left (a x\right )^{3} \,d x } \]

[In]

integrate(x^3*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^7 + 2*a^2*c^2*x^5 + c^2*x^3)*sqrt(a^2*c*x^2 + c)*arctan(a*x)^3, x)

Sympy [F]

\[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\int x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}\, dx \]

[In]

integrate(x**3*(a**2*c*x**2+c)**(5/2)*atan(a*x)**3,x)

[Out]

Integral(x**3*(c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**3, x)

Maxima [F]

\[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3} \arctan \left (a x\right )^{3} \,d x } \]

[In]

integrate(x^3*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x, algorithm="maxima")

[Out]

integrate((a^2*c*x^2 + c)^(5/2)*x^3*arctan(a*x)^3, x)

Giac [F(-2)]

Exception generated. \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\int x^3\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]

[In]

int(x^3*atan(a*x)^3*(c + a^2*c*x^2)^(5/2),x)

[Out]

int(x^3*atan(a*x)^3*(c + a^2*c*x^2)^(5/2), x)